Topological Field Theory, Higher Categories, and Their Applications

TL;DR

This work argues that Extended Topological Field Theory in dimensions beyond two is naturally governed by weak $n$-categories, with boundary conditions and defects furnishing higher morphisms. It analyzes the Rozansky-Witten 3D TFT to illustrate monoidal deformations of the 2-periodic derived category $D_{\mathbb Z_2}(Coh(Y))$ via Maurer–Cartan data, and it connects 4D GL-twisted gauge theories to Geometric Langlands through Montonen–Olive duality, emphasizing dimensional reduction to 2D brane categories such as $D^b_{G_{\mathbb C}}(Coh({\mathfrak g}[2]))$ and Fukaya categories. The paper argues that higher-categorical structures underpin observed dualities, offering a physical route to Langlands-type correspondences and suggesting a broader $(\infty,n)$-categorical formalism. It also highlights open questions about rigorous foundations, higher-dimensional boundary conditions, and the precise categorical equivalences predicted by dualities in this framework.

Abstract

It has been common wisdom among mathematicians that Extended Topological Field Theory in dimensions higher than two is naturally formulated in terms of n-categories with n> 1. Recently the physical meaning of these higher categorical structures has been recognized and concrete examples of Extended TFTs have been constructed. Some of these examples, like the Rozansky-Witten model, are of geometric nature, while others are related to representation theory. I outline two application of higher-dimensional TFTs. One is related to the problem of classifying monoidal deformations of the derived category of coherent sheaves, and the other one is geometric Langlands duality.

Figures (9)

• Figure 1: Morphisms in the category of boundary conditions are boundary-changing point operators.
• Figure 2: A boundary-changing point operator is equivalent to a (possibly nonlocal) boundary condition.
• Figure 3: Composition of morphisms corresponds to fusing boundary-changing point operators. Fusion product is denoted by a dot.
• Figure 4: Boundary-changing line operator ${\mathsf A}$ from a boundary condition ${\mathbb X}$ to a boundary condition ${\mathbb Y}$ and boundary-changing line operator ${\mathsf B}$ from ${\mathbb Y}$ to a ${\mathbb Z}$ can be fused to produce a boundary-changing line operator from ${\mathbb X}$ to ${\mathbb Z}$. This operation is denoted $\otimes$.
• Figure 5: Boundary-changing line operators between boundary conditions ${\mathbb X}$ and ${\mathbb Y}$ are objects of a category ${\mathsf W}_{{\mathbb X}{\mathbb Y}}$. A morphism ${\mathcal{O}}_{{\mathsf A}{\mathsf B}}$ from an object ${\mathsf A}$ to an object ${\mathsf B}$ is a point operator inserted at the junction of ${\mathsf A}$ and ${\mathsf B}$.

Theorems & Definitions (1)

• Conjecture